[Fluxus] commits: icosphere

[Claude Heiland-Allen]

(What follows is a bit off-topic, sorry..)

Kassen wrote:
> Can't we somehow abstract this stuff
> http://en.wikipedia.org/wiki/Polyhedroninto a equation that would get
> us all of the basic (platonic) ones and a
> good amount of the not so basic ones from a few numbers?

Good question, and the answer (perhaps surprisingly) is yes.

If you're interested in this kind of stuff, the book "Regular Polytopes" 
by HSM Coxeter is the one to get - it's got a *lot* of information in 
it, and not too pricy.

I have some Haskell code that follows Coxeter's equations for 3D and 4D, 
but it's quite a long process:
1. Essentially in 3D you give the Schlaefli symbol {p,q} which means 
there are q p-gons around each vertex (so {4,3} is a cube).
2. From that you calculate a fundamental (spherical) triangle
3. Then the whole symmetry group follows by reflections
4. Then you can pick a starting point inside the fundamental triangle
5. Then apply each of the group actions (as matrices)

And you end up with a sort of truncated regular polyhedron, depending 
where the point you picked is.

Points 1-3 can be done once for all time, the symmetry group never 
changes and there are a finite number of "interesting" groups.

This is the older version of the code that calculates the symmetry 
groups in 4D (need to push the newer versions...):
http://gitorious.org/polytopiary/reflex/blobs/master/Polychoron.hs

Some older info (need to update with newer info...):
http://claudiusmaximus.goto10.org/cm/2009-10-15_reflex_preview.html
http://claudiusmaximus.goto10.org/g/reflex/


Claude
-- 
http://claudiusmaximus.goto10.org